here is something that's been bothering me for a while: say I have a charged particle beam that can be described by one bivariate Gaussian in X dimension and another bivariate Gaussian in Y dimension. The parameters of the bivariate gaussian are the position (X) of an individual particle and the projection of the tangent of the angle onto the X axis (Xnu). The angle and position can be correlated, hence the covariance between X and Xnu may be not zero. The same for the Y axis: Y and Ynu. Each distribution is characterized by its own variance, so there are var(X), var(Xnu), var(Y), and var(Ynu) as well as cov(x,Xnu), cov(Y,Ynu). Covariance between X and Y is not accounted for but also not important in this problem.
Now, I would like to experimentally measure the beam spot size and I have two available techniques: (1) a 2D scintillation screen that gives me X,Y map of beam intensities in the plane perpendicular to XY (2) a pinhole detector that moves along X or Y to collect the beam profile. Neither of the two detectors gives the particle direction information.
In (1) I get the map of the intensities and fit 2D gaussian in X and Y to give me var(X) and var(Y) (actually I'm after beam spot sizes, so sqrt(var(X)) and sqrt(var(Y))). In (2) I get a profile that typically runs across the center of the spot along X and Y directions, so for Y=0 and X=0, respectively. These are the 'conditional' profiles and they should also give me the same values for var(X) and var(Y).
Should or should not?
I looked into a statistics textbook and found the proof that the variance of a projected bivariate gaussian (which is what I measure by fitting the 2D gaussian in (1)) and the conditional gaussian (which is what I get from my experiment in (2)) should be equal. I also found that in the beam optics physics literature there is a distinction between the 'conditional' (aka plane) and the 'projected' (aka space) beam spot standard deviations. The article that I've found says that the conditional beam spot size is sqrt(2) times bigger than the projected beam spot size (variance(conditional) = 2* variance (projected)).
I'm confused. Are the variances equal or are they not? Your input is appreciated.